SECTION VI. THE SIMPLE MECHANICAL POWERS. 45. THE object of all machines considered in a statical point of view is to enable a certain force as P, which is called the Power, to be in equilibrium with a second force, as W which is termed the Weight. If of these counteracting forces, P be smaller than W, the power is said to act at a mechanical advantage; if it be greater than W, at a mechanical disadvantage. These distinctive names are given to the two forces because we are most familiar with the use of machines when employed for the purpose of raising or moving heavy bodies by the application of a small force or power. 46. All machines or mechanical powers may be considered as being formed by combinations of cords and rods with hard surfaces. The simplest of them are the Lever, Wheel and Axle, Toothed Wheels, Pulley, Inclined Plane, Wedge, and Screw. The Lever. 47. DEF. The simple lever is a rigid rod without weight, capable of turning freely about some fixed point in its length, which is called the fulcrum. The power and weight are applied at two other points of the rod. The relative positions of these points with respect to the fulcrum suggest the division of levers in three classes as follows: Fig. (1). B Suppose C to be the fulcrum, A the point of application of the power which may be the tension of a cord produced by pulling by the hand; B that of the weight, as of a body; then fig. (1) PV W represents a lever of the first kind where the fulcrum is between the power and the weight. The Crowbar in some methods of use is an example of this species; Scissors and Carpenters' Pincers are double levers of the same kind, the joint being the fulcrum. Fig. (2). A B Fig. (2) represents a lever of the second kind, where the fulcrum is at one end, the power at the other, and the weight in the middle. A crowbar be used so as to be a lever of this kind. A cork-squeezer and an oar are other examples; in may the last, the blade of the oar in the W water is the fulcrum. Nut-crackers are double levers of the same kind. B Fig. (3). A Fig. (3) represents a lever of the third kind, where the fulcrum is again at one end of the rod, but the power is in the middle, and the weight at the other end. The bones of the arm where the muscle produces the power, are examples of this kind of lever; Blacksmiths' tongs, shears, &c. are double levers of the same sort. W 48. DEF. The distances AC, BC of the points of application of the power and weight from the fulcrum, are termed the arms of the lever; they need not be in the same straight line; when they are not so, the lever is said to be a bent lever. The directions of the power and weight may be any what ever. 49. In all cases the lever is kept in equilibrium by the forces called the power and weight respectively, and the reaction of the fulcrum; hence the resultant of the power and the weight must equal this reaction of the fulcrum, and its direction must pass through the fulcrum, which is insured (Art. 28) when the algebraical sum of their moments about the fulcrum vanishes: this condition is sometimes enunciated in the following form-the power and the weight are inversely proportional to the lengths of the perpendiculars drawn from the fulcrum upon their directions. As the reaction of the fulcrum, or its equivalent, the pressure upon the fulcrum, is equal to the resultant of the power and weight, it follows that when the power and weight are parallel, and act in the same direction, it is equal to their sum, and when they are parallel, but act in opposite directions, it is equal to their difference (Art. 23). Wheel and Axle. 50. The annexed figure represents a wheel and axle. HH' is a cylinder capable of turning freely about pivots CC' at its extremities, AB is a wheel firmly fixed to the cylinder HH' and having the same axis. The power P acts by means of a cord or rope which is wrapped round the circumference of the wheel; the weight W is applied in a similar manner by a rope which is wound round HH', but which runs in an opposite direction to that of P. By the aid of a figure representing a section of the machine, made perpendicular to the axis, it is not difficult to see that the power and weight respectively are always acting at the extremities of a N W Р Toothed Wheels. 51. Suppose two wheels, whose circumferences are indented with teeth, to be so placed that their teeth fit together, and their axes lie parallel to each other: if then one of them be made to revolve about its axis by a force applied to it, it will communicate motion to the other by means of the mutual pressure of the teeth: this tendency to produce motion may be counteracted by the action of a force applied to the second wheel; these two forces may be called the power and the weight of the system, the P and the W as in the lever. P and W may be applied to their respective wheels in an almost infinite number of ways; it is not uncommon for them to be so by means of a rope coiled round an axle to the wheel in each case; the system then becomes a combination of a pair of wheels and axles. If then r, R be the radii of P's wheel and axle respectively, and r'R' those of Q's, it is not difficult, by the aid of Art. 50, to see that P R If R = R', since the teeth are set at equal distances on each wheel, and their numbers on each are therefore inversely proportional to the radius, we observe that the above form becomes P number of teeth on P's wheel = W number of teeth on W's wheel' The Pulley. 52. The Pulley is a small wheel capable of turning about its axis, which is fixed in a framework called the block; according as the block is fixed or moveable, so is the pulley termed a fixed or moveable pulley. The object of the pulley is to change the direction of the tension of a cord which passes over it as either the edge of the pulley which the cord touches, or the axis of the pulley, is always supposed to be perfectly smooth, no other change is produced in the force thus transmitted; it remains of the same intensity at every point of the cord. Combinations of pulleys may be made in a great variety of ways; in all of them, a force acting upon the first string is made to sustain another which is applied by the aid of the last; these two forces constitute the power and the weight P and W. |